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1 # Pendulum: Comparing Harmonic Oscillator with 4th order Taylor approximation

3 ### Taylor 4 approximation: phi'' = - g/r * (phi - 1/6 * phi^3)

4 # fortunately, the components of orders 0, 2 and 4 are zero

6 coefficient.1 -> g/r

7 coefficient.2 -> 1/6 # set to 0,167 or 0 (for harmonic solution)

8 coefficient.3(-1) -> -phi0

10 iintegrate phi'' -> -phi'

11 iintegrate -phi' -> phi

12 IC: -phi0

14 multiply phi, phi -> phi^2

15 multiply phi^2, phi -> phi^3

16 invert phi^3 -> -phi^3

18 cmultiply 1/6, -phi^3 -> -1/6*phi^3

19 isum phi, -1/6*phi^3 -> -(phi-1/6*phi^3)

20 cmultiply -(phi-1/6*phi^3), g/r -> -g/r*(phi-1/6*phi^3)

21 assign -g/r*(phi-1/6*phi^3) -> phi''

23 output phi -> out.x

25 ### Harmonic oscillator: phi'' = - g/r * phi

26 coefficient.5 -> g/r_h # identical to g/r

27 coefficient.7(-1) -> -phi0_h # identical to -phi0

29 iintegrate phi_h'' -> -phi_h'

30 iintegrate -phi_h' -> phi_h

31 IC: -phi0_h

32 invert phi_h -> -phi_h

34 cmultiply -phi_h, g/r_h -> -g/r*phi_h

35 assign -g/r*phi_h -> phi_h''

37 output phi_h -> out.y